arXiv:2201.02612v3 [physics.flu-dyn] 26 Jul 2022

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Natural Convection Heat Transfer from an Isothermal Plate

Aubrey G. Jaffere-mail: [email protected]

Abstract

Natural convection heat transfer formulas which are accurate over a wide range of Rayleighnumbers (Ra) are known for vertical and downward-facing plates, but not for upward-facing plates.From the thermodynamic constraints on heat-engine efficiency, this investigation derives formulasfor natural convection flow and heat transfer from upward-facing, isothermal plates. The union offour peer-reviewed data-sets spanning 1 < Ra < 1012 has 5.4% root-mean-squared relative error(RMSRE) relative to this new heat transfer formula.

This novel approach derives a formula nearly identical to Churchill and Chu (1975) for verticalplates at 1 < Ra < 1012, and improves the Schulenberg (1985) formula for downward-facing platesfrom 4.6% RMSRE to 3.8% on four peer-reviewed data-sets spanning 106 < Ra < 1012.

The introduction of the harmonic mean as the characteristic-length metric for vertical anddownward-facing plates extends those rectangular plate formulas to other convex shapes, achieving3.8% RMSRE on vertical disk convection from Hassani and Hollands (1987) and 3.2% from Kobusand Wedekind (1995).

Building on the work of Fujii and Imura (1972) and Raithby and Hollands (1998), the threeorthogonal plate formulas are combined to calculate the heat transfer at any plate inclination,achieving 4.7% RMSRE on the inclined plate measurements from Fujii and Imura.

This research did not receive any specific grant from funding agencies in the public, commercial, ornot-for-profit sectors.

Table of contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Prior theoretical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. Experimental data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. Unenclosed heat-engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55. Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66. Combining heat transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77. Upward-facing circular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78. Upward-facing measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99. Vertical rectangular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010. Downward-facing rectangular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111. Self-obstruction of vertical and downward-facing plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212. Ra scaling factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313. Vertical measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414. Downward-facing measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415. Characteristic-length metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516. Downward-facing circular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517. Vertical circular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618. Upward-facing square plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1719. Vertical rectangular plate with side-walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1720. Upward-facing rectangular plate with side-walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821. Inclined plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1922. Inclined plate with side-walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2124. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2225. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2326. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1

1. Introduction

Natural convection is the flow caused by nonuniform density in a fluid. It is a fundamental process withapplications from engineering to geophysics.

When a stationary, immersed object changes temperature, nearby fluid can change density as it warmsor cools. Under the influence of gravity, density changes cause fluid to flow. The rates of fluid flow andheat transfer from the object grow until reaching a plateau. This investigation seeks to predict the overallsteady-state heat transfer rate from an external, flat, isothermal surface inclined at any angle.

An “external” plate is one that fluid can flow around freely, especially horizontally. If enclosed, theenclosure must have dimensions much larger than the heated or cooled surface. Natural convection in anenclosure of size comparable to the heated or cooled surface can organize into cells of Rayleigh-Benardconvection, which is not treated here.

The characteristic-length L is the length scale of a physical system. For many heat transfer processes,it is the volume-to-surface-area or area-to-perimeter ratio of the heated or cooled object. There are severalcharacteristic-length metrics used for natural convection, some of which are valid only for convex objects.This investigation focuses on flat plates with convex perimeters.

There are three topologies of convective flow from external, convex plates.For a horizontal plate with heated upper face, streamlines photographs in Fujii and Imura [1] show

natural convection pulling fluid horizontally from above the plate’s perimeter into a rising central plume.Figure 1, below, is a diagram of this upward-facing convection. Horizontal flow is nearly absent at theelevation of the dashed line.

Kitamura, Mitsuishi, Suzuki, and Kimura [2] shows top-views of plumes from heated rectangular plateswith aspect ratios between 1:1 and 8:1. The plates with high aspect ratios have a plume over the plate’smid-line parallel to the longer sides, but not as long.

The streamlines photograph of a vertical plate in Fujii and Imura [1] shows fluid being pulled horizontallybefore rising into a plume along the vertical plate.

Modeled on a streamlines photograph in Aihara, Yamada, and Endo [3], Figure 2 is a flow diagram fora horizontal plate with heated lower face. Unheated fluid below the plate flows horizontally inward. It risesa short distance, flows outward closely below the plate, and flows upward upon reaching the plate edge.

Figure 1 flow above a heated plate Figure 2 flow below a heated plate

There is a symmetry in external natural convection; a cooled plate induces downward flow instead ofupward flow. Flow from a cooled upper face is the mirror image of flow from a heated lower face. Flow froma cooled lower face is the mirror image of flow from a heated upper face.

Sublimation from an upper face is downward convection when the dissolved sublimate is denser thanthe fluid. The rest of this investigation assumes a heated plate.

An important aspect of all three flow topologies is that fluid is pulled horizontally before being heatedby the plate. Pulling horizontally requires less energy than pulling vertically because the latter does workagainst the gravitational force. Inadequate horizontal clearance around a plate can obstruct flow and reduceconvection and heat transfer.

In fluid mechanics, the convective heat transfer rate is represented by the average Nusselt number (Nu).The Rayleigh number (Ra) is the impetus to flow due to temperature difference and gravity. A fluid’s Prandtlnumber (Pr) is its momentum diffusivity per thermal diffusivity ratio. These three “variable groups” aredimensionless (measurement units of the constituent variables cancel each other).

The characteristic-length L scales Nu; Ra is scaled by L3; Pr is independent of L.

2

Formulas for heat transfer can apply to mass transfer via analogous variable groups, such as Schmidtnumber (Sc) and Pr. Figure 3 has Sherwood number (Sh) instead of Nu on its vertical axis.

Previous investigations [1, 2, 4, 5] assumed that natural convection heat transfer formulas would differsubstantially when the convection was turbulent versus laminar. For their upward-facing plate, Lloyd andMoran [4] reported that the transition from laminar to turbulent flow occurred at Ra ≈ 8 × 106. The linesthey fitted to their data at greater and lesser Ra were disjoint at Ra = 8 × 106. However, with their fitlines removed, if Ra ≈ 8× 106 represents a discontinuity, then it is one of several, and subsumed within thescatter of their measurements in Figure 3.

101

102

104

105

106

107

108

109

Ra = 8 × 106

Sh

erw

oo

d n

um

ber

S

h

Rayleigh number Ra

Lloyd and Moran 1974 − turbulent

Lloyd and Moran 1974 − laminar

Figure 3 upward convection heat transfer from horizontal plate

About their measurements of vertical and downward tilted plates, Fujii and Imura [1] wrote:“Though the boundary layer was not always laminar near the trailing edge for large Gr Pr

[= Ra] values, no influence of the flow regime on the data shown in [their] Fig. 6 is appreciable.”

Churchill and Chu [5] concludes that one of its equations“. . .based on the model of Churchill and Usagi [6] provides a good representation for the mean heattransfer for free convection from an isothermal vertical plate over a complete range of Ra and Prfrom 0 to ∞ even though it fails to indicate a discrete transition from laminar to turbulent flow.”

The lack of a significant transition between the rates of (mean) heat transfer in laminar and turbulentnatural convection indicates that some more basic principle organizes natural convection.

The fundamental laws of thermodynamics make no distinction between laminar and turbulent flows.Considering a small object inside a very tall column of fluid as a closed system, natural convection is aheat-engine which converts the temperature difference between the object and fluid into flow of that fluid.The heated object is the heat source; fluid far above the object is the heat sink.

This investigation parlays thermodynamic constraints on heat-engine efficiency into a novel theoryof steady-state natural convection heat transfer from external, isothermal plates, and compares it withmeasurements from eighteen data-sets from seven peer-reviewed articles.

3

2. Prior theoretical work

Renno and Ingersoll [7] relates the “convective available potential energy” (CAPE) of a planetary at-mosphere to heat-engine efficiency. Atmospheric convection is analyzed as a four phase cyclic heat-engine.They introduce “total convective available potential energy” (TCAPE) in terms of the reversible heat-engineefficiency limit η = ∆T/T . For dry air:

TCAPE ≈ η cp ∆T ∆T = T − T∞ (1)

where cp is the fluid specific heat (at constant pressure), T is the absolute temperature of the ground(heat-source), and T∞ is the upper atmosphere (heat-sink) absolute temperature.

CAPE = ηN cp ∆T , where ηN is the heat-engine efficiency of natural convection. Renno and Ingersollassert that TCAPE ≈ 2CAPE. Thus, ηN ≈ η/2.

The Goody [8] cyclic heat-engine analysis splits atmospheric convection into four phases, three of themreversible, and accounts for their energy flows and entropy.

T∞ T ηA ηN = η/2 = ∆T/[2T ]240 K 300 K 10.2% 10.0%260 K [?] 295 K 7.7% 5.90%250 K 295 K 7.7% 7.63%

Table 1 atmospheric convection efficiency limit

Table 1 shows the sink (T∞) and source (T ) temperatures, the efficiency limits from Goody (ηA), and ηNcalculated from the temperatures. In the first row, ηA matches ηN within 2%. In the second row, ηA is 30%larger than ηN . The third row changes T∞ from 260 K to 250 K, matching ηA to ηN within 1%. If “260 K”was a misprint, then both studies agree that ηN ≈ η/2.

For upward convection heat or mass transfer from a horizontal surface, prior works [1, 2, 4, 9, 10]propose constant coefficients fitted to fractional powers of Ra, spanning various Ra ranges. The goal of thisinvestigation being a comprehensive formula, the present theory will be compared with the measurementspresented in these works, not with their piece-wise power-law approximations.

The natural convection heat transfer formula developed by Churchill and Chu [5] for average Nusseltnumber (Nu) from a vertical rectangular isothermal plate of height L is:

Nu1/2

= 0.825 +0.387Ra1/6

[

1 + (0.492/Pr)9/16]8/27

1 ≤ Ra ≤ 1012 (2)

Schulenberg [11] derives a formula for convection below a level isothermal strip of width 2L. Proposedis a corrected1 formula and its equivalent, normalized so that Pr appears only in the denominator:

Nu =0.631Ra1/5 Pr1/5

[

1 + 1.56Pr3/5]1/3

=0.544Ra1/5

[

1 + (0.477/Pr)3/5]1/3

(4)

Schulenberg also gives a formula for downward convection from a level isothermal disk using its radiusas the characteristic-length. The expression on the right side is its equivalent normalized form:

Nur =0.705Ra1/5 Pr1/5

[

1 + 1.48Pr3/5]1/3

=0.619Ra1/5

[

1 + (0.520/Pr)3/5]1/3

(5)

1 The Schulenberg [11] heat transfer formula for an isothermal strip was:

Nu =0.571Ra1/5 Pr1/5

[

1 + 1.156Pr3/5]1/3

=0.544 3

√1.156Ra1/5 Pr1/5

[

1 + 1.156Pr3/5]1/3

(3)

1.156 is the only 4-digit coefficient in the paper’s isothermal plate correlations; the others have 3 significantdigits. Figure 5 of the present work compares the effective Ra scale factor of all four of these formulas; the“1.156 Schulenberg strip 1/Ξ×” trace is 40% lower than the others at Pr ≪ 1.

4

3. Experimental data sets

There are robust measurements of natural convection heat and mass transfer in the peer-reviewed literature.For upward natural convection from a horizontal surface, the union of the following three data-sets spans

1 < Ra < 1012:

Goldstein, Sparrow, and Jones [9] measured sublimation of solid naphthalene in air with 1 < Ra < 10000.Lloyd and Moran [4] measured electrochemical mass transfer with 26000 < Ra < 1.6× 109.Fujii and Imura [1] measured heat transfer above a heated plate in water with 108 < Ra < 1012.

For natural convection from vertical plates:

Churchill and Chu [5] compared vertical convection heat transfer formulas with data-sets in seven fluidsfrom thirteen studies, together spanning 1 < Ra < 1013.Fujii and Imura [1] measured heat transfer from a heated, vertical plate in water with 107 < Ra < 1011.Kobus and Wedekind [12] measured heat transfer from three sizes of heated thermistor disks in air with200 < Ra < 104.Kobus and Wedekind [12] includes measurements from Hassani and Hollands (1987) of a large thermistordisk with 1 < Ra < 3× 105.

For downward natural convection from a horizontal surface:

Fujii and Imura [1] measured heat transfer below a heated plate in water with 107 < Ra < 1012.Aihara et al [3] measured heat transfer from a heated rectangular plate in air.Faw and Dullforce [13] measured heat transfer from a heated disk in air.

Fujii and Imura made heat transfer measurements spanning 180◦ of plate angles at Ra ≈ 108 and Ra ≈ 1010.

4. Unenclosed heat-engine

Although most textbook heat-engine analyses are of cyclic heat-engines, a continuous processes can alsoconvert a temperature difference into mechanical work, which qualifies it as a heat-engine.

Consider a large vertical column of still, dry air having molar massM under the influence of gravitationalacceleration g. Initially, the air will be in equilibrium, with uniform absolute temperature T∞ and a pressureprofile P which decays exponentially with altitude z:

P (z) = P0 exp

(−z gM

RT∞

)

(6)

Where R is the universal gas constant, the ideal gas law finds the density ρ of a parcel of air:

ρ =M P

RT(7)

A heated parcel of volume V has density ρh = M P/[

[T∞ +∆T ]R]

. The buoyancy force on it is:

[ρ− ρh] g V =g V M P ∆T

RT∞ [T∞ +∆T ](8)

Where 0 < ∆T ≪ T∞, and cp is the specific heat (at constant pressure) of the fluid, ∆Q = cp ρ V ∆Tis the heat required to raise the parcel temperature from T∞ to T = T∞ +∆T . The force on the parcel is:

[ρ− ρh] g V =

[

M P

ρRT

]

g∆Q

cp T=

g∆Q

cp T=

g ρ V ∆T

T(9)

As it rises, the parcel’s state changes. Temperature, volume, and pressure are three variables havingtwo degrees of freedom from formula (7). For a large vertical column of still, dry air, Fermi [14] teaches:

“Since air is a poor conductor of heat, very little heat is transferred to or from the expandingair, so that we may consider the expansion as taking place adiabatically.”

5

Hence, the temperature of a parcel of dry air drops g/cp ≈ 9.8 K per kilometer of altitude gain. In acolumn having initially uniform temperature, a heated parcel will rise until its temperature drops to T∞.

From the conservation of mass, ρ V = ρ0 V0, where ρ0 and V0 are the density and volume at altitudez = 0. The maximum work W which can be extracted from a buoyant parcel is the integral of upward forceformula (9) with respect to altitude z above the heated plate:

W =

∫ ∆T cp/g

0

[∆T − z g/cp] g ρ0 V0

Tdz =

g∆Q

cp T

∆T cp2 g

=∆Q∆T

2T(10)

The thermodynamic efficiency (W/∆Q) of this ideal convection heat-engine will be the thermodynamicefficiency limit for external convection, ηN :

ηN =W

∆Q=

∆T

2T(11)

Note that ηN is 1/2 of the (Carnot) reversible heat-engine efficiency limit η = ∆T/T .This derivation was for adiabatic gases whose coefficient of thermal expansion β = 1/T . More generally:

ηN =β∆T

2(12)

The system being in continuous operation, instead of energies W and ∆Q, power fluxes (W/m2) are ofinterest. The powers per heated plate area are Ik for the kinetic flux of the fluid and Ip for the plate total,which is also the convective power flux. The thermodynamic efficiency of a steady-state convection processis Ik/Ip, which the second law of thermodynamics constrains so that:

IkIp

≤ ηN (13)

Additional fluid properties used in this investigation are thermal conductivity k, kinetic viscosity ν, andthermal diffusivity α = k/[ρ cp]. h is the average convective surface conductance, with units W/(m2 ·K).

5. Dimensional analysis

“Scalable” heat transfer equations relate named, dimensionless “variable groups”, which themselves are func-tions of variables and other variable groups. “Dimensional analysis” discovers these dimensionless variablegroups and their scalable relationships.

Nusselt’s dimensional analysis of natural convection (from Lienhard and Lienhard [15]) employs fourvariable groups: average Nusselt number Nu, Prandtl number Pr, Π3, and Π4.

Nu ≡ hL

k=

Ip L

∆T k, Pr ≡ ν

α, Π3 ≡ L3

ν2g =

L g

[ν/L]2, Π4 ≡ β∆T (14)

The Nu = Ip L/[∆T k] equivalence was added for this investigation.From these variable groups come the dimensionless Grashof number (Gr) and Rayleigh number (Ra):

Gr ≡ Π3 Π4 =β∆T g L3

ν2, Ra ≡ Gr Pr =

β∆T g L3

αν(15)

From formula (12) and Π4 ≡ β∆T from formula (14):

ηN =Π4

2(16)

Let Π5 be the heat transport capacity per kinetic energy ratio. Π5 increases with β, g, L, and Pr:

Π5 = β g LPr2/cp (17)

The denominator cp, canceling one of the two factors of cp in Pr2, makes Π5 dimensionless.Two variable groups with power flux units (W/m2) will prove useful:

Φp =k∆T

LΦk =

[ ν

L

]3

ρΠ4 Π5 (18)

6

6. Combining heat transfers

Conduction and convection are both heat transfer processes.There is an unnamed form which appears frequently in heat transfer formulas:

F p(ξ) = F p0 (ξ) + F p

∞(ξ) (19)

Churchill and Usagi [6] wrote that such formulas are “remarkably successful in correlating rates oftransfer for processes which vary uniformly between these limiting cases.”

A value of p > 1 models competitive processes; the combined transfer rate is between the larger con-stituent rate and their sum.

p = 1 models independent processes; the combined transfer rate is the sum of the constituent rates.

The Churchill and Chu formula (2) for vertical plates has the form of equation (19) with p = 1/2.Natural convection requires some conduction to heat the fluid. This is consistent with cooperating processeshaving 0 < p < 1; when both are transferring, the combination is larger than their sum.

With F0(ξ) ≥ 0 and F∞(ξ) ≥ 0, taking the pth root of both sides of equation (19) yields a vector-spacefunctional form known as the ℓp-norm, which is notated ‖F0 , F∞‖p:

‖F0 , F∞‖p = (|F0|p + |F∞|p)1/p (20)

7. Upward-facing circular plate

For a horizontal upward-facing plate, Figure 1 shows that natural convection pulls fluid from the edges intoa central plume. The characteristic-length should be a function of radial distance from the edges to thecenter. This is accomplished by using the area-to-perimeter ratio L∗ as the characteristic-length L.

Lloyd and Moran [4] measured upward convection from horizontal disks, rectangles, and right triangleshaving aspect ratios between 1:1 and 10:1. They wrote: “It is immediately obvious that within the scatter ofthe data, approximately ±5 percent, the data from all planforms are correlated through the use of L∗, . . . ”

Goldstein et al [9] found that L∗ correlated their measurements with aspect ratios between 1:1 and 7:1.

Consider a horizontal disk with its upper face, having area A, heated to T∞ + ∆T . Its L = L∗ is 1/2of its radius. The power flowing from an object into a stationary, uniform medium is q = S k∆T , where kis thermal conductivity and S is the conduction shape factor (having length unit). For one side of a disk,Incropera, DeWitt, Bergman, and Lavine [16] gives S = 2D (= 8L∗). Converting conduction power fluxIp = q/A into conduction Nusselt number Nu∗

0 = Nu from formula (14):

Ip A = Nu∗

0 Ak∆T

L∗= q = S k∆T Nu∗

0 =S L∗

A=

8L∗ L∗

π [2L∗]2=

2

π≈ 0.637 (21)

Fluid heated near the plate converts thermal energy into kinetic energy by accelerating upward. Fluidaccelerating upward spreads apart, pulling fluid horizontally to maintain its density. At some elevation zt,the fluid no longer accelerates upward (otherwise, its velocity would be unbounded) and the horizontal flowis negligible, which is marked by the dashed line in Figure 1.

An ideal turbine at elevation zt would capture the upward kinetic energy of the plume. The kineticpower through the aperture would be ρAuu2/2, where u is the plume upward velocity; its flux, ρ u3/2.

Vertical acceleration pulls fluid horizontally at elevations between 0 and zt. “Fig. 14(f)” of Fujii andImura [1] shows that horizontal velocities are fairly uniform within that span. The kinetic flux should beproportional to ρ u3/2 scaled by zt. zt grows with u, but shrinks with kinematic viscosity ν because ofviscous losses. u/ν has reciprocal length units, while ρ u3/2 already has power flux units. This suggestsscaling ρ u3/2 by a (dimensionless) Reynolds number Re = uL/ν, which is used extensively for modelingforced flows. Let Rei = uLi/ν, where Li is the average length of flow parallel to the plate. For theupward-facing plate, Li = 2L = 2L∗; hence Rei = 2Re, leading to a kinetic power flux Reρu3. From theΠ5 dimensional analysis, ReiΠ5 [ρ u

3/2] is the maximum heat flux which could be transported by the flow

7

induced by u. Multiplying this heat flux by Π4/2 yields the maximum kinetic flux Ik which could resultfrom natural convection:

Ik = Reiρ u3

2

Π4

2Π5 =

ρL

2 νu4 Π4 Π5 u =

[

2 ν IkρLΠ4Π5

]1/4

(22)

With upward convection pulling fluid horizontally from the disk’s perimeter, heat transfer near theperimeter is more flow-induced than it is conduction. If the flow were parallel, 1/2 of the plate area wouldbe considered flow-induced. If the flow were radial, 1/4 would be considered flow-induced. However, thesquare plate photographs in Kitamura et al [2] show plumes as a network of connected ridge segments, nota central cone. An intermediate allocation is needed. The geometric mean of 1/2 and 1/4 is

√

1/8. Hence,√

1/8 ≈ 0.354 of the plate is designated as flow-induced, [1−√

1/8] ≈ 0.646 of the plate as conduction.Heat transfer from the flow-induced part of the plate will be proportional to Nu∗

0, L∗, and formula (22) u

in the dimensionless expression Nu∗0 uL

∗/[√8 ν] = Nu∗

0 Re/√8. As cooperating processes, conductive and

flow-induced heat transfers combine using the ℓ1/2-norm. Solving for plate power flux Ip from formula (14):

Ip =k∆T

LNu∗

0

∥

∥

∥

∥

1− 1√8,Re√8

∥

∥

∥

∥

1/2

=k∆T

LNu∗

0

∥

∥

∥

∥

∥

1− 1√8,

L√8 ν

[

2 ν IkρLΠ4 Π5

]1/4∥

∥

∥

∥

∥

1/2

(23)

Assume Re ≫√8, so the 1−

√

1/8 term can be ignored. From definitions (18) collect Φk and Φp terms:

Ip = ΦpNu∗

0√8

[

2 IkΦk

]1/4

(24)

The upper bound for Ik can be found by combining Ip formula (24) with ηN formulas (13) and (16):

Ik ≤ Π4

2Ip =

Φp Π4 Nu∗0√

8

[

Ik8Φk

]1/4

(25)

Dividing both sides of formula (25) by I1/4k , then raising both sides to the 4/3 power, isolates Ik:

Ik ≤[

Φp Π4 Nu∗0√

8

]4/3 [1

8Φk

]1/3

= Φp

[

Nu∗0√8

]4/3 [Φp Π4

Φk

]1/3Π4

2(26)

In the absence of obstruction, Ik and Ip will increase to the maximum allowed by upper-bound for-mula (26). Substituting Ik from formula (26) into formula (24) yields the asymptotic formula for Ip:

Ip = Φp

[

Nu∗0√8

]4/3 [Φp Π4

Φk

]1/3

(27)

Both Ip and Ik have 3√

Φp Π4/Φk factors. How does Φp Π4/Φk relate to formula (15) Ra?

Φp Π4

Φk=

k∆T

ρL

L3

ν3β g L

cp

ν2

α2=

β∆T g L3

αν= Ra (28)

Ip = Φp

[

Nu∗0√8

]4/3

Ra1/3 = Φp Nu∗ Nu∗ =Nu∗

04/3

4Ra1/3 ≈ 0.137Ra1/3 (29)

Restoring the ℓ1/2-norm from equation (23) into equation (29) yields the comprehensive formula fornatural convection heat transfer from an external, horizontal plate’s isothermal upper face:

Nu∗ =

∥

∥

∥

∥

∥

Nu∗

0

[

1− 1√8

]

,Nu∗

04/3

4Ra1/3

∥

∥

∥

∥

∥

1/2

≈[

0.642 + 0.370Ra1/6]2

(30)

Nu∗ formula (30) assumes unobstructed flow. A completely unobstructed apparatus is difficult to build.Measurements smaller than Nu∗ are expected.

Measurement bias and uncertainty can result in values slightly larger than Nu∗.

8

8. Upward-facing measurements

100

101

102

103

100

101

102

103

104

105

106

107

108

109

1010

1011

1012

Nu*0 = 0.637Nu*(0) = 0.412

Ra = 26900

Ra = 8 × 106

aver

age

Nu

ssel

t n

um

ber

or

Sh

erw

oo

d n

um

ber

Rayleigh number Ra

present work Nu* + 5%

present work Nu*

Fujii and Imura 1972 − 30 cm × 15 cmFujii and Imura 1972 − 5 cm × 10 cmLloyd and Moran 1974 − turbulentLloyd and Moran 1974 − laminarGoldstein et al 1973

Figure 4 upward convection heat transfer from horizontal plate

source data-set Pr or Sc orientation formula RMSRE bias scatter countFujii and Imura [1] – 30 cm× 15 cm 5.0 upward (30) Nu∗ 12.0% −11.4% 4.0% 11Fujii and Imura [1] – 5 cm× 10 cm 5.0 upward (30) Nu∗ 5.0% −0.4% 5.0% 10Lloyd and Moran [4] – electrochemical 2200 upward (30) Nu∗ 4.9% +0.7% 4.8% 39Goldstein et al [9] – sublimation 2.50 upward (30) Nu∗ 7.2% −2.3% 6.8% 26

Table 2 upward convection heat transfer from horizontal plate

Lloyd and Moran [4] estimated 5% scatter for their data. Two “Lloyd and Moran 1974 – laminar”points at Ra ≈ 26900 have values 14% and 19% larger than Nu∗ in Figure 4.

Ra ≈ 26900 was the smallest Ra measured by Lloyd and Moran; the next smallest Ra = 171750 was 6.4times larger. Range extremes are often the most susceptible to measurement bias. Having excesses severaltimes larger than 5%, the points at Ra ≈ 26900 should be excluded as outliers.

Excluding the two largest and two smallest measurements relative to Nu∗, the Lloyd and Moran mea-surements have a 4.9% root-mean-squared relative error (RMSRE) from Nu∗. This is a close match spanningfour orders of magnitude of Ra which includes the laminar-turbulent transition at Ra ≈ 8× 106.

RMSRE gauges the fit of measurements g(Ra) to formula f(Ra), giving each measurement equal weight.The root-mean-squared error of g(Ra) relative to f(Ra) at n points Raj is:

√

√

√

√

1

n

n∑

j=1

∣

∣

∣

∣

g(Raj)

f(Raj)− 1

∣

∣

∣

∣

2

(31)

Table 2 also splits RMSRE into bias and scatter. The root-sum-squared of bias and scatter is RMSRE.

Note that Nu∗(0) < Nu∗0 in Figure 4. Nu∗ models convection; it does not extend to static conduction.

The Fujii and Imura 30 cm× 15 cm upward-facing data-set is revisited in Section 20.

9

9. Vertical rectangular plate

The vertical characteristic-length L′ = L is the plate’s height. Conduction constant Nu′0 will not depend on

the plate’s width. The asymptotic case is a strip, an infinitely wide rectangle.Conduction shape factors are not well-defined with unbounded source areas, but Nusselt numbers can

be. Fortunately, Nu′0 for a strip can be related to square plate Nu0. Incropera et al [16] gives a dimensionless

shape factor q∗SS = 0.932 for both faces of a rectangular plate. For one face of an L× L square plate:

AS = 2A = 2L2 LS =

√

AS

4 π=

L√2 π

q =q∗SS

2k∆T

A

LS= q∗SS k∆T L

√

π

2(32)

Nu0Ak∆T

L= q = q∗SS k∆T L

√

π

2Nu0 = q∗SS

√

π

2≈ 1.168 (33)

Strip conduction Nu′0 must distribute over one dimension (vertical) what square plate conduction Nu0

distributes over two:Nu′

0 = Nu02 = q∗SS

2 π

2≈ 1.363 (34)

Fluid is pulled horizontally before rising into a plume at the plate. The upward flow is parallel; plate areais treated as 1/2 flow-induced, 1/2 conduction. The average length of contact with the plate is L/2, resultingin the Ik factor Re/2 = uL/[2 ν]. Fluid heated by the plate accelerates upward along its surface. This reducesthe effective length of contact by 1/2, resulting in Re/4 as the heat transfer factor in formula (36). Thekinetic and plate power fluxes are:

Ik =Re

2

ρ u3

2

Π4

2Π5 =

L

ν

ρ u4

8Π4 Π5 u =

[

ν

L

8 IkρΠ4 Π5

]1/4

(35)

Ip =k∆T

LNu′

0

∥

∥

∥

∥

1

2,1

2

Re

4

∥

∥

∥

∥

1/2

=k∆T

LNu′

0

∥

∥

∥

∥

∥

1

2,

L

8 ν

[

ν

L

8 IkρΠ4 Π5

]1/4∥

∥

∥

∥

∥

1/2

(36)

Assume Re = uL/ν ≫ 1 and ignore the conduction term; collect Φp and Φk terms from definitions (18):

Ip =Φp

8Nu′

0

[

8 IkΦk

]1/4

(37)

The upper bound for Ik can be found by combining Ip formula (37) with ηN formulas (13) and (16):

Ik ≤ Π4

2Ip =

Φp Π4 Nu′0

16

[

8 IkΦk

]1/4

(38)

Dividing both sides of formula (38) by I1/4k , then raising both sides to the 4/3 power, isolates Ik:

Ik ≤[

Φp Π4 Nu′0

16

]4/3 [8

Φk

]1/3

= ΦpNu′

0

4/3

8 3√2

[

Φp Π4

Φk

]1/3Π4

2(39)

The plate partially obstructs flow; Nu′ will be an upper bound. Reduce to Ra using equation (28):

Ip ≤ 2

Π4

Ik = ΦpNu′

0

4/3

8 3√2

[

Φp Π4

Φk

]1/3

= ΦpNu′

0

4/3

8 3√2

Ra1/3 ≥ Φp Nu′ (40)

Nu′ ≤ Nu′0

4/3

8 3√2

Ra1/3 ≈ 0.150Ra1/3 (41)

Reintroduce the ℓ1/2-norm into formula (41):

Nu′ ≤∥

∥

∥

∥

∥

Nu′0

2,Nu′

0

4/3

8 3√2

Ra1/3

∥

∥

∥

∥

∥

1/2

≈[

0.826 + 0.387Ra1/6]2

(42)

For large Pr, the Churchill and Chu equation (2) reduces to Nu =[

0.825 + 0.387Ra1/6]2.

The Ra term’s denominator in equation (2) is always greater than 1; it can only reduce the magnitudeof Nu. Therefore, formula (2) satisfies Nu′ upper-bound formula (42).

10

10. Downward-facing rectangular plate

Figure 2 shows flow in two plumes on horizontally opposite sides of the plate; downward-facing characteristic-length LR is 1/2 of the shorter plate edge. Compared with the vertical convection strip, LR = L′/2 andNu0 = Nu′

0/2. Plate areas are treated as 1/2 flow-induced, 1/2 conduction.For upward-facing and vertical plates, the induced flow brings unheated fluid into contact with the

plate, which is responsible for amplifying convection. In contrast, fluid below the downward-facing plate iswarmed by conduction through the fluid above it. The outward creep immediately below the plate staysin contact until it reaches a plate edge. The fluid’s temperature profile differs little from static conduction.Thus, static and dynamic heat transfers are combined using the ℓ1-norm.

Figure 2 shows convective flow experiencing three 90◦ changes of direction. Two horizontal accelerationsand decelerations of flow introduce two factors of 2Re = 2 uL/ν into Ik. The short upward accelerationand deceleration of flow below the plate is the only such occurrence among the three plate orientations.It slightly opposes buoyant flow because fluid immediately below the plate is less dense than fluid movingupward to replace it. There being no appropriate vertical distance, Re is used for the third factor in Ik.

Ik = 4Re3ρ

2u3 Π4

2Π5 =

L3

ν32 ρ u6Π4 Π5 u =

[

ν3

L3

Ik2 ρΠ4Π5

]1/6

(43)

All of the lower face is in contact with horizontal flow; the heat transfer factor is 2Re:

Ip =k∆T

LNu0

[

1

2+

2Re

2

]

=k∆T

LNu0

[

1

2+

L

ν

[

ν3

L3

Ik2 ρΠ4Π5

]1/6]

(44)

Assume uL/ν ≫ 1 and ignore the conduction term; collect Φp and Φk terms; then solve for Ip:

Ip = Φp Nu0

[

Ip2Φk

]1/6

Ip = ΦpNu0

6/5

21/5Ra1/5 = Φp NuR (45)

The plate obstructs heated flow from rising; thus, NuR will be an upper bound. Solving equation (45)for NuR, restoring the ℓ1-norm (addition), and substituting Nu′

0/2 for Nu0:

NuR ≤ Nu′0

4+

Nu′0

6/5

27/5Ra1/5 ≈ 0.341 + 0.550Ra1/5 (46)

For large Pr, the Schulenberg strip convection formula (4) reduces to 0.544Ra1/5. The formula (46)coefficient 0.550 is only 1.1% larger than 0.544. Being greater than 1, the denominator of formula (4) canonly reduce the magnitude of Nu. Therefore, formula (4) satisfies NuR upper-bound formula (46).

These three derivations can be generalized to equation (48) via formulas (47). L is characteristic-length;Nu0 is static conduction; E is the count of 90◦ changes in direction of fluid flow; B is the sum of the meanlengths of flows parallel to the plate divided by L; C is the plate area fraction responsible for flow inducedheat transfer; D is the effective length of heat transfer contact with the plate divided by L; p is the ℓp-norm p.

Ik =BReEρ u3

2

Π4

2Π5 Ip =

k∆T

LNu0 ‖1− C,C DRe‖p (47)

Nu =

∥

∥

∥

∥

∥

Nu0 [1− C] ,2+E

√

[C DNu0]3+E 2

BRa

∥

∥

∥

∥

∥

p

(48)

orientation L (Ra ∝ L3) Nu0 E B C D pupward L∗ Nu∗

0 1 2 1/√8 1 1/2

vertical L′ Nu′0 1 1/2 1/2 1/4 1/2

downward LR(= L′/2) Nu′0/2 3 4 1/2 2 1Table 3 derivation parameters

11

11. Self-obstruction of vertical and downward-facing plates

Defined in formula (14), the Prandtl number (Pr) is the momentum diffusivity per thermal diffusivity ratioof a fluid. Heat transfer from plates to fluids with small Pr is primarily conduction. Temperature changesin fluids with large Pr cause changes in density which induce fluid flow. The Nu′ and NuR upper-boundformulas (42) and (46) are asymptotic for large Pr.

Other than as a factor of Ra, Pr does not affect upward-facing heat transfer because the heated fluidflows directly upward, as does conducted heat. When heated fluid must take longer paths around self-obstructing vertical and downward-facing plates, its heat transfer potential is reduced.

Ra scales with L3; Pr is a property of 3-dimensional fluids. A function of Pr having values between 0and 1 should scale Ra in the vertical and downward-facing formulas. Both Schulenberg [11] and Churchilland Chu [5] realized their formulas’ dependence on Pr in this way, demonstrated as follows:

Expressing the denominator of vertical formula (2) as the sixth root of an ℓp-norm expression namedΞ′(Pr) in formula (49):

[

1 + (0.492/Pr)9/16

]8/27

=

[

∥

∥

∥

∥

1 ,0.492

Pr

∥

∥

∥

∥

9/16

]1/6

= [Ξ′(Pr)]1/6

(49)

Scaling Ra by 1/Ξ′(Pr) in the vertical upper-bound formula (42) makes it equivalent to formula (2):

Nu1/2

= 0.826 + 0.387

[

Ra

Ξ′(Pr)

]1/6

Ξ′(Pr) =

∥

∥

∥

∥

1 ,0.492

Pr

∥

∥

∥

∥

9/16

(50)

Similar treatment of the Schulenberg downward-facing strip and disk formulas (4) and (5) yields:

Nu = 0.544

[

Ra

ΞR(Pr)

]1/5

ΞR(Pr) =

∥

∥

∥

∥

1 ,0.477

Pr

∥

∥

∥

∥

3/5

(51)

Nu = 0.619

[

Ra

Ξr(Pr)

]1/5

Ξr(Pr) =

∥

∥

∥

∥

1 ,0.520

Pr

∥

∥

∥

∥

3/5

(52)

Figure 5 shows that, as Pr grows, the 1/Ξ(Pr) functions approach 1; as Pr shrinks, they approach 0.

The functions Ξ′(Pr), ΞR(Pr), and Ξr(Pr) are quite similar. Coefficients 0.492, 0.477, and 0.520 areall within 5% of 1/2. This suggests using 1/2 as the coefficient in a unified function Ξ∀(Pr). The ℓp-norm pparameters 3/5 = 0.6 and 9/16 = 0.5625 differ by less than 7%; and

√

1/3 ≈ 0.577 lies between them:

Ξ∀(Pr) =

∥

∥

∥

∥

1 ,0.5

Pr

∥

∥

∥

∥√1/3

(53)

Incorporating Ξ∀(Pr) into Nu′ and NuR upper-bound formulas (42) and (46) creates comprehensivevertical and downward convection formulas (54) and (55):

Nu′ =

∥

∥

∥

∥

∥

Nu′0

2,Nu′

0

4/3

8 3√2

[

Ra

Ξ∀(Pr)

]1/3∥

∥

∥

∥

∥

1/2

≈∥

∥

∥

∥

∥

0.682 , 0.150

[

Ra

Ξ∀(Pr)

]1/3∥

∥

∥

∥

∥

1/2

(54)

NuR =

∥

∥

∥

∥

∥

Nu′0

4,Nu′

0

6/5

27/5

[

Ra

Ξ∀(Pr)

]1/5∥

∥

∥

∥

∥

1

≈ 0.341 + 0.550

[

Ra

Ξ∀(Pr)

]1/5

(55)

12

12. Ra scaling factors

10−2

10−1

100

10−2

10−1

100

101

102

103

Ra

sca

lin

g f

acto

r

Prandtl number Pr

1.56 Schulenberg strip 1/ ΞR

Schulenberg disk 1/ Ξr

present work 1/ Ξ∀Churchill and Chu 1975 1/ Ξ′ 1.156 Schulenberg strip 1/ Ξ×

(two) Fujii and Imura 1972Goldstein and Lau 1983Faw and Dullforce 1982(two) Aihara et al 1972

Churchill and Chu 1975

0.3

0.4

0.5

0.6

0.7

0.7 1.0 1.5 2.0 2.5 3.0 4.0 5.0

Figure 5 Ra scaling functions

Figure 5 presents the Ra scaling functions 1/Ξ(Pr) used in the vertical and downward-facing formulas.Also shown are 1/Ξ(Pr) values calculated from coefficients fitted to measurements in the cited sources.

Table 4 lists each fitted value and its deviation from each Ra scaling function at the given Pr. Churchilland Chu 1/Ξ′ is closest to the fitted values.

If a candidate formula is correct, negative deviations of fitted (aggregate) values can result from flowobstructions or measurement bias; positive deviations can result only from measurement bias.

The positive 1/Ξ′ deviations, +2.5% and +2.9%, would indict two of the five sources in Table 4.The positive 1/Ξ∀ deviations, +0.5% and +0.6%, are tolerable.With values between those of 1/ΞR and 1/Ξ′, 1/Ξ∀ is the most plausible of these Ra scaling functions.

orientation source Pr 1/Ξ from fit 1/ΞR 1/Ξr 1/Ξ∀ 1/Ξ′ 1/Ξ×

vertical Fujii and Imura [1] 5.00 0.669 -3.7% -2.0% +0.5% +2.5% +7.5%downward Fujii and Imura [1] 5.00 0.652 -6.1% -4.5% -2.1% -0.1% +4.8%downward Goldstein and Lau [10] 2.50 0.565 -4.5% -2.2% +0.6% +2.9% +10.9%downward Faw and Dullforce [13] 0.71 0.347 -8.6% -5.0% -2.4% -0.0% +15.9%downward Aihara et al [3] 0.71 0.339 -10.7% -7.2% -4.7% -2.3% +13.3%downward Aihara et al [3] 0.71 0.310 -18.4% -15.2% -12.8% -10.7% +3.6%vertical Churchill and Chu [5] 0.024 0.036 -7.9% -0.8% -1.5% -0.9% +42.4%

Table 4 Ra scaling factors

The “1.156 Schulenberg strip 1/Ξ×” curve in Figure 5 is the Ra scaling function from formula (3).Section 2 argues that this formula from Schulenberg [11] contained a typographical error. Being 40% less thanthe other curves at Pr ≪ 1 corresponds to formula (3) taking values 10% less than (corrected) formula (4).

13

13. Vertical measurements

100

101

102

103

100

101

102

103

104

105

106

107

108

109

1010

1011

1012

1013

Nu ′(0) = 0.682

aver

age

Nu

ssel

t n

um

ber

N

u

Rayleigh number Ra

Pr = 0.70; present work Nu ′

Pr = 0.70 (air); KingPr = 0.70 (air); JakobPr = 0.70 (air); Cheesewright

Pr = 0.024; present work Nu ′

Pr = 0.024 (mercury); Saunders

Figure 6 heat transfer from vertical rectangular plate

source data-set Pr orientation formula RMSRE bias scatter countChurchill and Chu [5] – King 0.70 vertical (54) Nu′ 13.5% +11.1% 7.6% 8Churchill and Chu [5] – Jakob 0.70 vertical (54) Nu′ 4.7% +3.1% 3.5% 5Churchill and Chu [5] – Cheesewright 0.70 vertical (54) Nu′ 16.4% −15.4% 5.6% 6Churchill and Chu [5] – Saunders 0.024 vertical (54) Nu′ 5.3% −1.2% 5.1% 18

Table 5 heat transfer from vertical rectangular plate

Figure 6 and Table 5 present four data-sets from Churchill and Chu [5]. Differences between RMSREcomputed from Nu′ formula (54) and Churchill and Chu formula (2) are less than 0.6%.

14. Downward-facing measurements

8 10

15

20

30

40

60

80100

150

200

106

107

108

109

1010

1011

1012

aver

age

Nu

ssel

t n

um

ber

N

u

Rayleigh number Ra

Pr = 5.0 (water); present work Nu

RPr = 5.0; Fujii and Imura 1972 − 30 cm × 15 cmPr = 5.0; Fujii and Imura 1972 − 5 cm × 10 cm

Pr = 0.71 (air); present work Nu

RPr = 0.71; Aihara et al 1972 − 25 cm × 35 cmPr = 0.71; Faw and Dullforce 1982 − 18 cm diskPr = 0.71; 1.56 Schulenberg strip 1985

Figure 7 downward convection heat transfer from horizontal plate

source data-set Pr orientation formula RMSRE bias scatter countFujii and Imura [1] – 30 cm× 15 cm 5.0 downward (55) NuR 2.7% −0.8% 2.6% 8Fujii and Imura [1] – 5 cm× 10 cm 5.0 downward (55) NuR 4.2% −3.4% 2.5% 15Aihara et al [3] – 25 cm× 35 cm 0.71 downward (55) NuR 3.8% −3.7% 0.9% 2Faw and Dullforce [13] – 18.1 cm disk 0.71 downward (55) NuR 3.7% +1.8% 3.2% 3

Table 6 downward convection heat transfer from horizontal plate

Figure 7 and Table 6 present four downward-facing data-sets, NuR formula (55), and the correctedSchulenberg formula (4). Lack of a conduction term causes the Schulenberg trace to be less than NuR.

14

15. Characteristic-length metrics

For upward convection, L∗ (area-to-perimeter ratio) is well-defined for any flat, convex plate.The formulas for vertical (54) and downward convection (55) were developed for rectangular plates.

More general characteristic-length metrics are needed to model heat transfer from other plate shapes.The next sections develop new characteristic-length metrics for vertical and downward-facing plates,

and re-derive the formulas for Nu∗ (30), Nu′ (54), and NuR (55) from alternative plate shapes.

16. Downward-facing circular plate

Schulenberg equations (4) and (5) for downward-facing strips and disks have matching exponents, but theircoefficients and characteristic-lengths differ: LR is 1/2 of the rectangle’s shorter side, versus disk radius R.Can NuR formula (55) predict heat transfer for both shapes using a single characteristic-length metric?

Figure 2 shows fluid closest to the heated surface flowing outward from the mid-line. If the plate edgeis at varying distances from the mid-line (in the direction of flow), then some sort of length averaging isneeded. Flow will be faster over the shorter distances because it experiences less drag; this suggests use ofthe “harmonic mean”, in which small values have more influence than large.

For rectangles and disks, the mid-line is one of the plate’s equal-area bisectors. For rectangles it isparallel to the longer sides; but it is not the longest bisector, which is a diagonal. Being parallel to thelonger sides implies that the mid-line is perpendicular to the shorter sides. It will also be perpendicular tothe shortest equal-area bisector; and this works for disks as well, where all diameters are bisectors.

Consider a flat heated surface with its convex perimeter defined by functions y+(x) > 0 and y−(x) < 0within the range −R < x < R along the equal-area bisector which is perpendicular to the shortest equal-areabisector. Let LR be the combined harmonic mean of |y+(x)| and |y−(x)|:

LR = 1

/

∫ R

−R

[

1

|y+(x)|+

1

|y−(x)|

]

dx

4R(56)

For rectangular plates, LR formula (56) is 1/2 of the shorter side’s length, the same characteristic-lengthused by Schulenberg [11]. For disks of radius R:

y+(x) = −y−(x) =√

R2 − x2 LR = 4R

/

∫ R

−R

2√R2 − x2

dx =2

πR (57)

Schulenberg’s disk formula (5) used radius R as the disk’s characteristic-length. Converting the Nur

coefficient from characteristic-length R to LR is 0.619 [2/π]1/5 ≈ 0.566, which is within 3% of the 0.550coefficient of NuR formula (55).

The derivation of downward formulas (46, 55) was for a square plate. To derive the downward formulafor a disk, Nu0 will be recalculated; it will be scaled larger because of the increased flow over the shorterdistances. Recalling from Section 7, the conduction shape factor for one side of a disk is S = 2D = 4R.

Nu0 =S LR

A=

4RLR

π R2=

8

π2≈ 0.811 (58)

Upward-facing disk Nu∗0 was scaled by

√

1/8 in formula (27) because its flow was midway betweenparallel and radially inward; downward-facing Nu′

0/2 was unscaled in (rectangle) formula (44). The flowfrom a downward-facing disk spreads from a diameter; an intermediate scale is needed. The geometric meanof 1 (unscaled) and

√

1/8 is 4√

1/8. Scaling Nu0 by the reciprocal, 4√8 ≈ 1.682, yields the rectangular Nu′

0 ≈1.363. With the LR harmonic mean metric (56) and 4

√8Nu0 = Nu′

0, the rest of the derivation is unchanged.Thus, downward formula (55) works for both rectangular and circular plates.

Figure 7 and Table 6 include three disk measurements from Faw and Dullforce [13].4√8Nu0 = Nu′

0 suggests an exact expression for the rectangular plate’s dimensionless shape factor q∗SS :

q∗SS2 π

2=

8 4√8

π2q∗SS

2 =16 4

√8

π3q∗SS =

4 8√8

π3/2≈ 0.93158+ (59)

15

17. Vertical circular plate

Consider a flat vertical plate as an array of narrow vertical plates. With Ra = 0, integrate h across thevertical slices of heights L(x); then solve h = Nu′

0 k/L′ for L′:

Nu′0 k

L′= h =

1

2R

∫ R

−R

Nu′0 k

L(x)dx L′ = 2R

/

∫ R

−R

1

L(x)dx (60)

Therefore, the vertical characteristic-length L′ is the harmonic mean of vertical spans L(x).For a rectangle of height L, L′ = L. For the disk of radius R, L′ = 4R/π. Recalculating Nu0:

Nu0 =S L′

A=

4RL′

π R2=

16

π2≈ 1.621 (61)

Instead of the bifurcated flow from the downward-facing plate, flow is along full vertical spans of thedisk; 1/2 of the diameter-to-circle fringing which scaled downward-facing disk Nu0 by 4

√8 should apply to

vertical disks. With this scaling, Nu04√8/2 = Nu0/

4√2 = Nu′

0 ≈ 1.363, and the rest of the derivation isunchanged. Thus, vertical formula (54) works for disks and axis-aligned rectangular plates using L′ harmonicmean formula (60).

Kobus and Wedekind [12] measured natural convection heat transfer from vertical thermistor disks inair. They also included a series of similar measurements from Hassani and Hollands (1987). Pr was notspecified; 0.71 is assumed. Each disk’s diameter d and thickness t are specified in Figure 8.

Air heated by the two disk sides flows over the upper half of the rim, inhibiting upper rim heat transfer.The lower half of the rim transfers heat at the same per area rate as the sides. The disk effective surfacearea, including half of the rim, is π d2/2 + π d t/2. Being normalized for area, Nu′ should be scaled by:

π d2/2

π d2/2 + π d t/2=

d2

d2 + d2 [t/d]=

1

1 + t/d(62)

To convert characteristic-lengths to the harmonic mean, Kobus and Wedekind Ra gets scaled by [2/π]3;Nu gets scaled by [2/π] and formula (62). Hassani and Hollands used different characteristic-lengths for Raand Nu; their Ra gets scaled by 1/[2

√π]3, while Nu gets scaled by

√π/4 and formula (62).

1

2

4

6

810

20

100

101

102

103

104

105

aver

age

Nu

ssel

t n

um

ber

N

u

Rayleigh number Ra

present work Nu ′ + 5%

present work Nu ′

Hassani and Hollands d = 82.0 mm; t/d = 0.1Kobus and Wedekind d = 19.99 mm; t/d = 0.069Kobus and Wedekind d = 15.29 mm; t/d = 0.058Kobus and Wedekind d = 7.43 mm; t/d = 0.155 air, Pr = 0.71

Figure 8 heat transfer from vertical disk

source data-set Pr orientation formula RMSRE bias scatter countHassani and Hollands [12] – 82 mm 0.71 vertical disk (54) Nu′ 3.8% −3.4% 1.6% 26Kobus and Wedekind [12] – three sizes 0.71 vertical disk (54) Nu′ 3.2% −0.4% 3.1% 19

Table 7 heat transfer from vertical disk

Figure 8 and Table 7 present the vertical disk heat transfer measurements.In the clever design using thermistors, only horizontal clearance and the wires attached to the disk

centers remained as obstructions, achieving a close match with the present theory in Table 7.

16

18. Upward-facing square plate

Converting square plate Nu0 formula (33) from L′ to L∗ is division by 4. Expanding q∗SS from equation (59):

Nu0

4=

q∗SS

4

√

π

2=

8√8√2π

=85/8

4 π≈ 0.292 (63)

The square plate’s flow will be moderately radial, scaling midway between the reciprocal of√

1/8 from

the upward-facing disk, and 4√8 from the downward-facing disk’s bifurcated flow. The geometric mean of

√8

and 4√8 is 83/8 ≈ 2.181. Scaling Nu0/4 by 83/8 yields 2/π = Nu∗

0 ≈ 0.637 from equation (21). The rest ofthe derivation is unchanged. Thus, upward convection Nu∗ formula (30) works for disks and square plates.

19. Vertical rectangular plate with side-walls

The Fujii and Imura [1] apparatus had (unheated) perpendicular side-walls creating a channel from the plate.In Table 8, the vertical 30 cm and 5 cm plates average 44% and 18% less heat transfer than expected

by Nu′ vertical formula (54). Clearly, formula (54) is incorrect for vertical plates with side-walls.Formula (64) incorporates an additional factor of Re in Ik to model the side-wall drag. The side-walls

obstruct horizontal flow, so the combined length of flow along the plate and side-wall is 2L; heat transfercontact along the plate is L. The ℓ1/2-norm changes to the ℓ1-norm

Ik = 2ReReρ u3

2

Π4

2Π5 =

L2

ν2ρ u5

2Π4 Π5

u =

[

ν2

L2

2 IkρΠ4 Π5

]1/5

(64)

Ip =k∆T

LNu′

0

∥

∥

∥

∥

1

2,1

2Re

∥

∥

∥

∥

1

=k∆T

LNu′

0

[

1

2+

L

2 ν

[

ν2

L2

2 IkρΠ4 Π5

]1/5]

(65)

Using the general formula (48) with E = 2, B = 2, and D = 1:

Nu′w =

Nu′0

2+

Nu′0

5/4

25/4

[

Ra

Ξ∀(Pr)

]1/4

(66)

100

101

102

103

101

102

103

104

105

106

107

108

109

1010

1011

aver

age

Nu

ssel

t n

um

ber

N

u

Rayleigh number Ra

water, Pr = 5.0

vertical Nu ′

vertical with side walls Nu′

w

Fujii and Imura (1972): 0.56 Ra1/4

Fujii and Imura (1972): 30cm plate

Fujii and Imura (1972): 5cm plate

Figure 9 vertical Fujii and Imura plates

source data-set Pr orientation formula RMSRE bias scatter countFujii and Imura [1] – 30 cm× 15 cm 5.0 vertical (54) Nu′ 43.8% −43.7% 3.0% 5Fujii and Imura [1] – 30 cm× 15 cm 5.0 vertical (66) Nu′

w 2.2% −0.5% 2.2% 5Fujii and Imura [1] – 5 cm× 10 cm 5.0 vertical (54) Nu′ 18.9% −18.3% 4.7% 6Fujii and Imura [1] – 5 cm× 10 cm 5.0 vertical (66) Nu′

w 5.2% +5.0% 1.6% 6Table 8 vertical Fujii and Imura plates

Table 8 details the performance of formula (66) for vertical plates with side-walls. RMSRE has decreasedsubstantially to 2.2% and 5.2%. Figure 9 shows Nu′

w ≤ Nu′, as required by efficiency constraints.

17

20. Upward-facing rectangular plate with side-walls

The L∗w area-to-perimeter ratio for side-walled upward-facing plates excludes side-wall length L′ from theperimeter length (2w + 2L′) because there is no flow through side-walls. Thus, L∗w = L′ w/[2w] = L′/2.

In Table 2 (and Table 9), the Fujii and Imura data-sets averaged 0.4% and 11.4% less than expectedfrom Nu∗ formula (30). The 30 cm plate having side-walls twice as long as its 15 cm channel width madeits flow similar to vertical plate flow in each half of the plate towards the center-line. L = L∗w = L′/2.

Fluid rises after heating; so the Ik and heat transfer factors are both Re/2. As with the side-walledvertical plate, conduction and flow-induced heat transfer combine using the ℓ1-norm (addition):

Ik =Re

2

ρ u3

2

Π4

2Π5 =

L

ν

ρ u4

8Π4 Π5

u =

[

ν

L

8 IkρΠ4 Π5

]1/4

(67)

Ip =k∆T

L

Nu′0

2

∥

∥

∥

∥

1

2,1

2

Re

2

∥

∥

∥

∥

1

=k∆T

LNu′

0

∥

∥

∥

∥

∥

1

4,

L

8 ν

[

ν

L

8 IkρΠ4 Π5

]1/4∥

∥

∥

∥

∥

1

(68)

Flow is partially obstructed by the side walls; hence, Ra will be scaled by 1/Ξ(Pr). Using the generalformula (48) with Nu0 = Nu′

0/2, B = 1/2, C = 1/2, D = 1/2, and p = 1:

Nuw =Nu′

0

4+

Nu′0

4/3

8 3√2

[

Ra′

Ξ(Pr)

]1/3

(69)

Note that, unlike Nu∗ formula (30), Nuw formula (69) depends on Ξ(Pr).

102

103

107

108

109

1010

1011

1012

water, Pr = 5.0

aver

age

Nu

ssel

t n

um

ber

N

u

Rayleigh number Ra

upward Nu*

upward convection with side−walls Nu

w

Fujii and Imura 1972 − 30 cm × 15 cm

Fujii and Imura 1972 − 5 cm × 10 cm

Figure 10 upward-facing Fujii and Imura plates

source data-set Pr orientation formula RMSRE bias scatter countFujii and Imura [1] – 30 cm× 15 cm 5.0 upward (30) Nu∗ 12.0% −11.4% 4.0% 11Fujii and Imura [1] – 30 cm× 15 cm 5.0 upward (69) Nuw 6.0% −1.8% 5.7% 11Fujii and Imura [1] – 5 cm× 10 cm 5.0 upward (30) Nu∗ 5.0% −0.4% 5.0% 10Fujii and Imura [1] – 5 cm× 10 cm 5.0 upward (69) Nuw 18.2% +17.7% 4.1% 10

Table 9 upward-facing Fujii and Imura plates

Figure 10 and Table 9 show that Nuw formula (69) is effective for the 30 cm plate, but not for the 5 cmplate. The 5 cm plate’s 10 cm channel width is twice the 5 cm side-wall length. This makes its flow moreradial than parallel. Thus, Nu∗ formula (30) is more appropriate for the 5 cm plate.

orientation L (Ra ∝ L3) Nu0 E B C D pupward L∗w = L′/2 Nu′

0/2 1 1/2 1/2 1/2 1vertical L′ Nu′

0 2 2 1/2 1 1Table 10 side-walled plate parameters

Table 10 lists the general derivation formula(48) parameters for the side-wall flow topologies.

18

21. Inclined plate

Let θ be the angle of a plate from vertical. Face up θ = −90◦; vertical θ = 0◦; face down θ = +90◦. Thisinvestigation has derived and tested formulas for rectangular and circular plates at −90◦, 0◦, and +90◦.

Following Fujii and Imura [1], the Ra′ argument to vertical formula (54) gets scaled by |cos θ| to modelthe reduced vertical convection from an inclined plate.2 Following Raithby and Hollands [17], the Ra∗

and RaR arguments to the upward and downward formulas (30) and (55) get scaled by |sin θ|.Raithby and Hollands take the maximum convective surface conductance h = kNu/L, not Nu, to

determine the overall convection. This is because Nu′, Nu∗, and NuR have different characteristic-lengths,while h is independent of L. Taking the maximum asserts that the associated flow topologies are mutuallyexclusive. This extreme competition is the ℓ∞-norm, which is equivalent to the max( ) function:

h = k

{

max(

Nu′(|cos θ|Ra′)/L′, NuR (|sin θ|RaR) /L′)

if ∆T sin θ ≥ 0;

max(

Nu′(|cos θ|Ra′)/L′, Nu∗(|sin θ|Ra∗)/L∗)

if ∆T sin θ ≤ 0;(70)

Rayleigh numbers can be expressed in terms of (vertical plate) Ra′:

h = k max

(

Nu′(|cos θ|Ra′)

L′,1

LRNuR

(

|sin θ|Ra′[

LR

L′

]3))

∆T sin θ ≥ 0 (71)

h = k max

(

Nu′(|cos θ|Ra′)

L′,1

L∗Nu∗

(

|sin θ|Ra′[

L∗

L′

]3))

∆T sin θ ≤ 0 (72)

The downward-facing topology flows outward from opposite plate edges; there is no plate area availablefor the vertical flow topology. Thus, Nu′ and NuR are mutually exclusive in formula (71). The upward-facingtopology draws fluid from the whole perimeter. Thus, Nu′ and Nu∗ are mutually exclusive in formula (72).Note that formula (72) mutual exclusion may not hold when part of the perimeter flow is obstructed.

Ideally, when Ra = 0, h should be independent of θ. For a 1 m square plate in k = 1 fluid, h(0◦) =h(+90◦) = Nu0

′/2 ≈ 0.682, but h(−90◦) ≈ 1.646. When θ = −90◦ forces Ra′ cos θ to 0, only the conductionterm remains. As noted in Section 8, Nu∗ formula (30) does not extend to static conduction.

Nu∗ and Nu′ track measurements well near Ra ≈ 1 in Figures 4, 6, and 8. Ignoring Nu∗ when Ra′ sin θ >−[L∗/L′]3, and NuR when Ra′ sin θ < [LR/L

′]3, avoids the conduction term competition at θ ≈ 0:

h = k

max(

Nu′(|cos θ|Ra′)/L′, Nu∗(

|sin θ|Ra′ [L∗/L′]3)

/L∗)

if Ra′ sin θ < −[L∗/L′]3;

max(

Nu′(|cos θ|Ra′)/L′, NuR

(

|sin θ|Ra′ [LR/L′]3)

/LR

)

if Ra′ sin θ > [LR/L′]3;

Nu′(

|cos θ|Ra′)

/L′ otherwise.

(73)

Note that L∗/L′ ≤ 1/2 and LR/L′ ≤ 1/2 are true for any flat, convex plate face.

For Ra > 1, the proposed h formula (73) will match non-side-walled horizontal and vertical platemeasurements to their appropriate k Nu∗/L∗, k Nu′/L′, and k NuR/LR values.

22. Inclined plate with side-walls

The Fujii and Imura [1] apparatus had side-walls. In Table 6, NuR formula (55) has less than 5% RMSRE;it is used as the side-walled downward-facing formula with LR = L′/2, regardless of which side is shorter.

To adapt h formulas (71) and (72) to apparatus with side-walls, Nu′w formula (66) replaces Nu′. This

leads to proposed downward h formula (74) for side-walled plates:

h = k max

(

Nu′w(|cos θ|Ra′)

L′,NuR

(

|sin θ|Ra′/23)

L′/2

)

∆T sin θ ≥ 0 (74)

Fujii and Imura photographs show the plume originating in the middle of the 5 cm plate when θ = −90◦.The origin shifts 9% toward the elevated end of the plate when θ = −85◦. The plume for the 30 cm plateat θ = −60◦ originates in the upper 1/4 of the plate. Plume movement with θ indicates that regions ofupward-facing and vertical convection shared the side-walled plate in the Fujii and Imura apparatus.

2 Fujii and Imura [1] credits B. R. Rich with the idea of scaling vertical Ra by | cos θ|. It can be thoughtof as a reduction in the effective gravitational acceleration g, which scales Ra linearly.

19

The side-walled upward and vertical topologies compete for horizontal flow in the channel. If competitionwere between perpendicular flows, they would combine as the root-sum-squared, which is the ℓ2-norm. Tocompete for horizontal channel flow, two changes in direction are required; they combine as the ℓ4-norm.

The 30 cm plate has side-walls twice as long as its channel width w. Flow through this channel will beprimarily parallel, as modeled by Nuw formula (69). The proposed upward h formula for the 30 cm plate is:

h = k

∥

∥

∥

∥

∥

Nu′w(|cos θ|Ra′)

L′,Nuw

(

|sin θ|Ra′/23)

L′/2

∥

∥

∥

∥

∥

4

∆T sin θ ≤ 0 w < L′ (75)

The 5 cm plate’s 10 cm channel is twice as wide as its length; horizontal flow will be more radial thanparallel. At θ = −90◦ it is modeled by Nu∗ formula (30), but with characteristic-length L∗w = L′/2.

Fujii and Imura streamlines photographs show the vertical Nu′w and upward-facing Nuw flow modes

having uniform rates of horizontal flow between the lower heated plate edge and pause elevation zt.Uniform horizontal flow is not the case for the 5 cm plate at θ = −45◦. The horizontal flow is slower

near the lower plate edge and increases with elevation. The plume lacks a clear origin. It does not matchany flow topology described thus far.

The upward-facing flow topology has bilateral symmetry; there is no flow between the halves created bysevering along a plane of symmetry. Thus, half of the upward-facing flow topology is also a flow topology.Its general formula (48) parameters are the same as the upward-facing topology, except that L = 2L∗. Theθ = −45◦ flow topology is modeled as the ℓ4-norm of h′

w and h∗/2, where h∗/2 is the heat transfer from thishalf upward flow topology. The proposed upward h formula for the 5 cm plate is:

h = k

∥

∥

∥

∥

∥

Nu′w(|cos θ|Ra′)

L′,Nu∗

(

|sin θ|Ra′/23)

L′/γ(θ)

∥

∥

∥

∥

∥

4

∆T sin θ ≤ 0 (76)

γ(θ) = max (1,min (2, | tan θ|+ 1− w/L′)) w > L′ (77)

At θ = −90◦, heat transfer is h∗; so γ(−90◦) = 2. At θ = −45◦, it is∥

∥h′w , h∗/2

∥

∥

4; so γ(−45◦) = 1.

The transition between γ = 2 and γ = 1 depends on θ, w, and L′. Dimensional analysis yielding formula (77)localizes the transition to w/L < |tan θ| < w/L+ 1, whose bounds are marked by arrows in Figure 11.

100

200

300

400

500600700800900

−80 −60 −40 −20 +0 +20 +40 +60 +80

water, Pr = 5.0; k = 0.60

− | tan θ | = w / L; θ = −63.4−−− | tan θ | = w / L + 1; θ = −71.6

mea

n s

urf

ace

con

du

ctan

ce,

− h,

W/(

m2 K

)

plate angle from vertical θ, degrees

5 cm × 10 cm; Ra′ = 108

30 cm × 15 cm; Ra′ = 10

10

Fujii and Imura 1972100

200

300

400

500

+76 +78 +80 +82 +84 +86 +88 +90

mea

n s

urf

ace

con

du

ctan

ce,

− h,

W/(

m2 K

)

plate angle from vertical θ, degrees

water, Pr = 5.0; k = 0.60

5 cm × 10 cm; Ra′ = 108

30 cm × 15 cm; Ra′ = 10

10

Fujii and Imura 1972

Figure 11 inclined Fujii and Imura plates Figure 12 inclined plate detail

source data-set Pr orientation formula RMSRE bias scatter countFujii and Imura [1] – 5 cm× 10 cm 5.0 inclined (74, 76) h 5.8% −2.4% 5.3% 15Fujii and Imura [1] – 30 cm× 15 cm 5.0 inclined (74, 75) h 3.3% −2.2% 2.6% 14

Table 11 inclined plate heat transfer

Tables 6, 8, 9, and 11 show that the present theory is sufficient to explain, with RMSRE between 2.2%and 6.0%, the Fujii and Imura heat transfer measurements of horizontal, vertical, and inclined plates.

20

23. Discussion

Renno and Ingersoll [7] and Goody [8] found that the heat-engine efficiency limit for atmospheric convectionis 1/2 of the reversible heat-engine efficiency limit η. This investigation finds that η/2 is the limit for externalnatural convection generally. Reversible heat engines, such as Stirling engines, can be more efficient than η/2.External convection is not reversible.

The Fujii and Imura [1] side-walled plates do not qualify as external because fluid was not free to flowhorizontally near the plate. The side-wall formulas are not general, particularly around w ≈ L′. Theywere investigated primarily to gauge how well the combination of ℓp-norm with trigonometric scaling of Raexplains heat transfer from inclined plates.

Evidence from Lloyd and Moran [4], and statements in Fujii and Imura [1] and Churchill and Chu [5],that the laminar-turbulent transition was irrelevant to natural convection heat transfer were published inthe early 1970s. Yet the belief that it governs external plate natural convection has persisted [10, 17, 2].

Much of the subsequent natural convection literature investigates local flow properties. Such studies donot inform this investigation’s systemic-invariants analysis. However, streamlines photographs were crucialto characterizing the flow topologies and deriving the present formulas.

The pause in horizontal flow above the upward-facing 5 cm plate is visible in the θ = −90◦ photographfrom Fujii and Imura [1]. Their photographs of vertical and downward-facing plate streamlines do not includeenough of the space above the plates to see the horizontal pause expected by this investigation.

The harmonic mean integral in formulas (56) and (60) converges only when the perimeter curve isperpendicular to the integration axis at both integration limits. For downward-facing plates this includes allcircles, ellipses, and rectangles; for vertical plates this includes all circles, ellipses, and trapezoids (including

rectangles) with two vertical edges. For3√Ra′ ≫ 1, vertical plate Nu′(L′) ∝ L′. Hence, the heat flow rate

h′ = kNu′/L′ is sensitive to L′ only at small Ra′ values.

All of the heat transfer measurements were digitized from graphs in the cited works using the “Engauge”software. Measurements obscured by other points in the graph were excluded.

source data-set Pr or Sc orientation formula RMSRE bias scatter countGoldstein et al [9] – sublimation 2.50 upward (30) Nu∗ 7.2% −2.3% 6.8% 26Fujii and Imura [1] – 30 cm× 15 cm 5.0 upward (69) Nuw 6.0% −1.8% 5.7% 11Fujii and Imura [1] – 5 cm× 10 cm 5.0 upward (30) Nu∗ 5.0% −0.4% 5.0% 10Lloyd and Moran [4] – electrochemical 2200 upward (30) Nu∗ 4.9% +0.7% 4.8% 39

Churchill and Chu [5] – Cheesewright 0.70 vertical (54) Nu′ 16.4% −15.4% 5.6% 6Churchill and Chu [5] – King 0.70 vertical (54) Nu′ 13.5% +11.1% 7.6% 8Churchill and Chu [5] – Saunders 0.024 vertical (54) Nu′ 5.3% −1.2% 5.1% 18Fujii and Imura [1] – 5 cm× 10 cm 5.0 vertical (66) Nu′

w 5.2% +5.0% 1.6% 6Churchill and Chu [5] – Jakob 0.70 vertical (54) Nu′ 4.7% +3.1% 3.5% 5Hassani and Hollands [12] – 82 mm 0.71 vertical disk (54) Nu′ 3.8% −3.4% 1.6% 26Kobus and Wedekind [12] – three sizes 0.71 vertical disk (54) Nu′ 3.2% −0.4% 3.1% 19Fujii and Imura [1] – 30 cm× 15 cm 5.0 vertical (66) Nu′

w 2.2% −0.5% 2.2% 5

Fujii and Imura [1] – 5 cm× 10 cm 5.0 downward (55) NuR 4.2% −3.4% 2.5% 15Aihara et al [3] – 25 cm× 35 cm 0.71 downward (55) NuR 3.8% −3.7% 0.9% 2Faw and Dullforce [13] – 18.1 cm disk 0.71 downward (55) NuR 3.7% +1.8% 3.2% 3Fujii and Imura [1] – 30 cm× 15 cm 5.0 downward (55) NuR 2.7% −0.8% 2.6% 8

Fujii and Imura [1] – 5 cm× 10 cm 5.0 inclined (74, 76) h 5.8% −2.4% 5.3% 15Fujii and Imura [1] – 30 cm× 15 cm 5.0 inclined (74, 75) h 3.3% −2.2% 2.6% 14

Table 12 measurements versus present theory

Table 12 summarizes statistics for the eighteen data-sets presented in Figures 4, 6, 7, 8, 9, 10, 11, andtheir associated tables. They are grouped by orientation and ordered by decreasing error relative to thepresent work, All but three of these data-sets have RMSRE of 6% or less, quantitatively supporting thepresent theory for horizontal, vertical, and inclined plates.

21

24. Conclusions

Streamline photographs, dimensional analysis, and the thermodynamic constraints on heat-engine efficiencycombine to give a theoretical derivation of a comprehensive heat transfer formula for upward convection froman external, isothermal flat plate with convex perimeter:

Nu∗(Ra∗) = Nu∗

0

∥

∥

∥

∥

∥

1− 1√8,Nu∗

01/3

4Ra1/3

∥

∥

∥

∥

∥

1/2

Ra∗ > 1

‖F0 , F1‖p = (|F0|p + |F1|p)1/p Nu∗

0 =2

π≈ 0.637

Applying the same technique to vertical and downward-facing plates derives heat transfer upper bounds.Modeling the convection reduction due to self-obstruction as a Ra scaling factor 1/Ξ(Pr) yields the com-prehensive formulas for vertical Nu′ and downward NuR:

Nu′(Ra′) =

∥

∥

∥

∥

∥

Nu′0

2,Nu′

0

4/3

8 3√2

[

Ra′

Ξ(Pr)

]1/3∥

∥

∥

∥

∥

1/2

Ra′ > 1

NuR(RaR) =Nu′

0

4+

Nu′0

6/5

27/5

[

RaRΞ(Pr)

]1/5

RaR > 1

Ξ(Pr) =

∥

∥

∥

∥

1 ,0.5

Pr

∥

∥

∥

∥√1/3

Nu′

0 =85/4

π2≈ 1.363

The reduction in effective gravitational acceleration g is also a Ra scaling factor. An external, isothermalplate inclined at angle θ from vertical has average convective surface conductance:

h = k

max(

Nu′(|cos θ|Ra′)/L′, Nu∗(

|sin θ|Ra′ [L∗/L′]3)

/L∗)

if Ra′ sin θ < −[L∗/L′]3;

max(

Nu′(|cos θ|Ra′)/L′, NuR

(

|sin θ|Ra′ [LR/L′]3)

/LR

)

if Ra′ sin θ > [LR/L′]3;

Nu′(

|cos θ|Ra′)

/L′ otherwise.

Ra′ is the Rayleigh number computed with vertical characteristic-length L′.The upward characteristic-length L∗ is the area-to-perimeter ratio.The vertical characteristic-length L′ is the harmonic mean of the perimeter vertical spans.The downward characteristic-length LR is the harmonic mean of the perimeter distances to that bisectorwhich is perpendicular to the shortest bisector.

The harmonic mean metrics extend vertical Nu′ and downward-facing NuR to non-rectangular plates.

The present theory makes no distinction between laminar and turbulent flow.The present theory was compared with eighteen data-sets from seven peer-reviewed articles, testing

circular, rectangular, and inclined rectangular plates, laminar and turbulent flows, with 0.024 < Pr < 2200and 1 < Ra < 1012. All but three of the data-sets had between 2% and 6% RMSRE from the present theory.

The Nu∗ formula improves accuracy and Ra range substantially over the piece-wise power-laws currentlyemployed for predicting upward convection heat transfer.With less than 1% difference between Nu′ and the Churchill and Chu (1975) vertical formula, there islittle need to replace it in existing applications.However, the published Schulenberg (1985) formula returns values significantly smaller than NuR, whichshould replace it.

22

25. Nomenclature

A = plate area (m2)cp = fluid specific heat at constant pressure (J/(kg ·K))g = gravitational acceleration (m/s2)h = average convective surface conductance (W/(m2 ·K))

Ik, Ip = kinetic, plate power flux (W/m2)k = fluid thermal conductivity (W/(m ·K))L = characteristic-length (m)M = air molar mass (kg)

Nu0, Nu = conduction, average Nusselt numberP = air pressure (N/m2)Pr = Prandtl numberq = conduction power (W)

q∗SS = dimensionless conduction shape factorR = disk radius (m)R = universal gas constant (J/(kg ·K))

Ra, Re = Rayleigh number, Reynolds numberS = conduction shape factor (m)T = temperature (K)u = fluid velocity (m/s)V = air volume (m3)W = work (J)w = distance between side-walls (m)

y+(x), y−(x) = perimeter functions (m)

Greek Symbols

∆Q = heat (J)∆T = T − T∞ = temperature difference (K)

α = k/[ρ cp] = fluid thermal diffusivity (m2/s)β = fluid thermal expansion coefficient (K−1)η = thermodynamic heat-engine efficiencyν = fluid kinematic viscosity (m2/s)

Π3, Π4, Π5 = dimensionless variable groupsΦk, Φp = kinetic, plate power flux (W/m2)

ρ = fluid density (kg/m3)θ = surface angle from vertical (−90◦ is face up)

Ξ(Pr) = Ra self-obstruction factor

Superscripts and Subscripts

[ ]∗ = upward-facing plate[ ]′ = vertical plate[ ]0 = conduction[ ]∀ = unified[ ]A = atmospheric natural convective[ ]i = induced flow along plate[ ]k = kinetic[ ]N = natural convective[ ]p = plate[ ]R = downward-facing plate[ ]r = downward-facing disk[ ]S = dimensionless shape factor[ ]× = misprinted formula[ ]w = with side-walls[ ]∞ = bulk fluid

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Acknowledgments

Thanks to Dave Custer, Rich Hilliard, Roberta Jaffer, and anonymous reviewers for their useful suggestions.

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